My friends and I are perplexed. How does the fact that 95% of the human body is water affect buoyancy?
What are you perplexed by? Affects the buoyancy in what way?
First: Buoyancy is the upward force exerted on an object by the fluid around it. Archimede’s Principle states that “An object in a fluid is buoyed up by a force equal to the weight of the fluid displaced.”
So, if the body were all water, in water, it would be in equilibrium; held up by the surrounding water. Other components in the body affect the density and weight of the body compared to the water, making it more or less buoyant. Personally, I’ve found that the greatest influence on human buoyancy is changing the volume of the body by breathing. Most people, I’m sure, have floated in a pool and noticed that as they breathe out, they have a tendency to sink, and vice-versa. Obviously, the gas in the lungs and tissues create a condition where the weight of the body(and it’s density) are overcome by the bouyancy of gas in water, since I don’t think(?) any tissues or fluids in the body are less dense than normal H2O.
Does that make sense? 
With a lung full of air people float as their density is less than that of water.
When drownings occur the lungs are usually filled with water and anything else that is around (leaves, mud, sand) as the person attempts to inhale while under water, the body looses it’s natural boyancy and sinks.
Rather unplesantly the body of the deceased will then float again after three or four days. I assume that this is due the natural decay of the body (bacterial action producing CO2 and decreasing the bodies density again???).
In theory this should make fat-bottomed people float more easily face down, while pot-bellied people are more likely to float on their backs.
Most critters are in a fairly interesting place, density-wise. Earth creatures are composed of large amounts of stuff with the same density as water (water, blood, and the “wettest” tissues), a lot of stuff slightly more dense than water (bone and perhaps cartliage?), and a lot of stuff slightly less dense than water (fat and perhaps lung tissue?)
Stunned fish float to the surface, unless they are sharks, in which case they sink. But both are close to water in density, which is how they are able to control depth. Humans do the same; even though are lungs are a small part of our volume, we are so close to water in density that we can shift from positive buoyancy to negative buoyancy just by breathing. Compare that to a duck, whose feather seal in a lot of air so it can float efforlessly, but must kick constantly to stay underwater.
Fish control their boyancy using a swim bladder, an air filled organ which can be inflated or deflated in order to maintain a neutral boyancy.
Sharks don’t have a swim bladder and thus must keep moving if they don’t want to sink as their density is heavier than water.
And no one has mentioned the fact yet that the OP is simply incorrect in its suggestion that the body is 95% water when it’s actually more along the order of 70%?
Regarding sharks and their buoyancy;having seen at least two species of shark hovering without moving forward, I have to say that some are not negatively buoyant. I have read they have very large oil filled livers that give them positive buoyancy.